 So the first session will begin with a talk by Professor Manaswita Bose. She is a faculty member in the energy science and engineering department of IIT Bombay. And she is also a co-PI, I mean co-principal investigator of the open form project of FOSI IIT Bombay. So over to you Manaswita ma'am, you can begin the session. And the talk will be on laminar flow through a pipe. So over to you ma'am. Good morning everyone. You are familiar with laminar flow through a pipe and perhaps also channel. I'll take you through these example problems highlighting some salient features. Needless to say that you can stop me at any point for discussion wherever you have any doubt, just stop me. Now the learning objectives are three. One is to connect to the basic form of the conservation laws to discuss laminar flow through channel and pipe. And the reason I have intentionally taken both problems channel and pipe is pictorial representation, schematic representation for both are exactly the same. However, they're different and I wanted to highlight at this point. I wanted to highlight this point. And then we will discuss a sample problem using a simple wage mesh. These are the tasks for this session. Now, when it comes to our mind laminar flow to be precise, low RE flow. And any idea why I said it is low RE flow to be precise. Because it is laminar flow ma'am. Yes. So there is a low Reynolds number flow. So we are saying a low RE. Okay. So what happens if the Reynolds number is infinite? Suppose if an ideal flow, ideal fluid flows through a pipe, will the flow be laminar? Yes, an ideal fluid flow will be laminar. So that's why these equations are to be precise. These equations are for low RE flow. Yes. So when it comes to our mind that the low RE flow, the equations that this is these equations are through pipe. The actual velocity is a function of radial position and that is a representation of fully developed flow. That is the velocity does not change with actual position. Maximum velocity is given minus dp dz. The minus sign of course takes care of the negative pressure gradient. The flow is driven by the pressure gradient and R squared by 4 mu. Delta p is 32 mu Lv by d squared and delta p can be rewritten in form of a couple of non-dimensional numbers and where f is the friction factor which is 64 by RE. This is the set of equations that comes to our mind. Exact schematic representation for a channel flow. However, the equations change. Here x is the cross flow direction. I should have drawn it. This is the cross flow direction. So use it is the flow direct z is the flow direction. Use it as a function of x. You max is one minus x squared by h squared h is the half channel width. You max is given as minus partial p partial z h squared by two mu. Here you max is three by two that is one one point five one and a half. You average. So the difference here it was. You average is twice you max here you. Sorry. You max is twice you average here you max is one and a half times of you average delta p is two mu L you max by eight square. If you write in terms of you average it is 12 you mu 12 mu L you average by a square a is the total channel which is the half channel. So there is a difference and why is this difference what is this difference due to the difference is due to the RDR effect. Okay, as we move from the as we move along the radial direction the area increases in case of pi because of this RDR effect. Though the schematic representation is exactly the same planar view is exactly the same equations are different. And that we can see from the Cartesian form of the governing equation and the governing equations in the polar coordinate and that's the connect I wanted to make the basic form of the equations. First thing the basic form of equations that we saw is mass and momentum conservation equation and the assumptions that we make is fluid is a continuum and I'm sure you would have discussed all these things in yesterday's yesterday's session. So I will very quickly go past it. Okay, when we say fluid is a continuum, what what decides whether fluid is a continuum or not it is nuts and number which is defined as the ratio of the mean free path to the system, mean free path to the system lens scale and mean free path to the system lens scale has to be less than one. What does it mean? Suppose we considered in the flow domain we consider a tiny volume. What is this time, what does this tiny volume mean, the volume, which is a point refer refers to an infinitesimal volume. And the average field variables are determined. However, this point contains this infinitesimal volume contains a large number of molecules and this is a small exercise I always like to do to give an idea. I do not know whether you have come across this exercise before but if not then maybe it's a good idea to take half a minute or so off break to do this exercise. So if you have an imaginary cube of one micron aside, okay, cube of one micron aside, and at normal temperature and pressure, which is 273 Kelvin and one atmospheric pressure. Can you say how many molecules of ideal gas is there in the volume? A quick calculation, we know that one mole is contained in 22.4 liter, right? One mole occupies 22.4 liter and one mole is Avogadro number. Can we do this quick calculation? I don't know, you want the accurate number but just an order of magnitude value. What kind of number we get? Are we doing these calculations? I think when it is 10 to the power 16. 10 to the power 16, one micron cube and 22.4 liter meaning one liter is one decimeter cube. Okay, if we do these calculations, we see that this cube contents tend to be roughly about 10 to the power 7 molecules, order of magnitude analysis. Now, if a small volume contains this many molecules, so we have enough number of molecules to calculate the average property. So these points represent average properties, averaged over what? Average over the molecules, right? We get the field variables which we use in all our macroscopic equations that is the field variables which pressure, temperature, density, velocity etc. And how else we can see? Suppose when we measure the properties. Suppose if we had the sensor probe which the probe was of the size of the mean free path, then we would see the fluctuations we just seen here. Okay, but if the probe is of the size of larger than the mean free path, these fluctuations wouldn't be seen. So when if we are in this region where the molecular fluctuations are not seen, we can say that we are in the continuum regime and that means that the number is much less than 1. This is the formal definition of continuum. We always say that fluid is a continuum. Okay, now when we have established fluid is a continuum, then the other thing we have to do is we have to write down the mass and momentum conservation equation for fluid. Now mass and momentum conservation equation we have been writing for ages for a system which is like DMDT of system is zero. This is the total derivative is the ordinary derivation. This says that mass cannot be created or destroyed. And this is Newton's second law of motion where we say that rate of change of momentum is the sum of rate of change of momentum of the system is some of the forces acting on the system. These two things we know, but we have to write them in terms of the, we have to write them for fluid. And when we say we are writing it for fluid, we write it for the control volume and this is how when we write the ordinary derivative for the system on the control volume, this is the conversion. This is the transformation. The ordinary derivative, there is a partial derivative and there is a flux term. I am not going through the derivation, but I am going to give you the physical interpretation. Again, you may have seen this in the previous lectures. I'll go through it very quickly. The partial derivative term tells you that if we focus at the control volume and this is Eulerian approach, so if you focus at the control volume, what is happening at the control volume with respect to time alone. That is the definition of the partial derivative and the flux term tells you that if we focus at the control volume, what is the net influx, what is the net inflow of the property. So, you know, we are talking about the mass and the momentum. Yes. In the Reynolds transportation theorem, this is the Reynolds transportation theorem, right? Absolutely. So, what is the system here? It is the control volume which is the system or something else is the system. What is the system in this Reynolds transportation theorem? No, see, the system is a defined mass. So, how are we going to define the mass in the control volume? Just give me one second. Can you see my screen now? Yes. So, I will define the system and control volume. So, system, you see this dark region? Yes. Okay, this is system. System is a quantity of mass, a region in space which you have selected for the study. This is the system. Control volume is something which is a properly selected region in space. So, this circuiting you are seeing here? Yes. Where I am using my cursor, this is the control volume. So, system is a defined mass. This one is the system which can move, okay, which is flowing, right? You see that this is flowing? So, at some point of time, say it was somewhere here, then at time P it came and occupied the control volume and then it has left and it is going out. It has partially left and then at some other time it has completely left and it has gone out. This is the relationship between system and control volume. So, my confusion is in the continuous flow when there is a continuous flow. My confusion is in the continuous flow, can I visualize the system? There is a continuous flow. We cannot see it. You imagine you are sitting in front of a continuous flow. Okay, some open channel flow or some flow through a pipe where you can see. Okay. Now, you imagine you have used the dye and you have dyed some fluid element that you can track. Okay, that becomes your system. Now, you are focusing at some point. So, that system that dyed particles, dyed fluid particles, some occupy that your control volume and leaves. You can continuously see. You can visualize it. You know, you sit in front of a river. You see that. Okay. You can check in the lab. You know, if you have a transparent pipe, you can visualize it. Right. And this is precisely what is the depiction of Reynolds transport theorem that it comes occupies at some time t is equal to zero. It has occupied at time t plus delta t it has left. So, it has left some volume in between and it has occupied some new volume. And this is that, you know, that dn dt where n is the generalized property dn dt of the system when we calculate this is what is the physical description of the Reynolds transport theorem. Okay. Okay. Thank you. Now, this mass rate of change of masses of the system is zero. And this is the this is what when we write in on the control volume. And here is the momentum conservation. So, rate of change of momentum on the rate of change of momentum of the control volume is the sum of the forces acting on the control volume. These are the two equations that we are going to deal with. Rate of change of mass of the control volume is zero rate of change of momentum of the control volume is the sum of the forces acting on the control. So, these are the two equations, but this is the integral form. We have to use the differential form for that we are using we are taking making use of the Gauss divergence theorem. And one more thing we have to do we have to write force in terms of the stress surface force we have to write in terms of the stress. This is a second order tensor, which again I think you have gone through in the previous lecture so I will go through it in detail. Now, after we do a little bit of algebra, we get the we get the familiar form of the mass and the momentum conservation equation. We still have kept the shear stress down. Here we will be using the constitutive relationship, which is for the Newtonian fluid. And for incompressible fluid, we will be having that divergence of u is zero will set this to zero. So, shear stress is mu into grad u plus grad u transpose. Now, what is incompressible fluid incompressible fluid we say we define in terms of Mach number, which is the ratio of the velocity of the fluid to the velocity of sound in the same medium. This is important in the same medium. And if it is if my Mach number is less than 0.3, it is a convention less than 0.3. We say we will be considering incompressible that is change in density with respect to pressure is negligible. Okay, this is the interpretation with that interpretation we go ahead and if we use this if we substitute this expression for tau we get this particular equation rho partial u partial t plus rho u dot grad u is equal to minus grad p plus mu del square u plus rho g. And this is the form of the equation that you know. So, for incompressible equation we have del dot u is zero and mu del square u these two equation form the complete set of equation. Of course, because we have this differential equations nonlinear and second order, we need boundary conditions and there are different types of boundary condition you can define the value of the variable itself. That means we can define the value of the velocity self at the wall, we can define the gradient at the wall and for heat transfer of course this is not for heat transfer there is another different another boundary condition will not discuss that. Now coming to the, this is the background, the background of the problem. Now coming to the user point of view, how do we define a problem? Now we will define a problem and we will solve it. Okay, so the problem definition is simulate the velocity profile of an incompressible Newtonian fluid flowing at a steady state to a horizontal pipe with circular cross section. Okay, if this is the problem statement that is given to you, what are the input, what are the information that one should know, we should know the pi diameter length properties of the fluid and velocity and the volume or the volumetric flow rate of the fluid something we should know. Okay, now the properties of the fluid. Okay, now one more thing I should have written here for analytical solution if we need to get in to get to any analytical solution, we should make one assumption here that the fluid is that the flow is fully developed. And that's what I have written here, see when we are writing the boundary conditions here we said okay at the wall it is no slip condition that means fluid assumes the velocity of the solid at the wall and we have written the boundary condition at the inlet and the outlet inlet we have said for analytical solution if you want to get an analytical solution because unless we simplify the equation we cannot get an analytical solution. So for analytical solution at the inlet also we assume that it is fully developed flow. So what does what does fully developed flow mean? Fully developed flow means yes. So velocity profile will not change further means it will remain same. Right. Absolutely. So that means suppose if this is the z direction that means duz dz is going to be zero. Yes. Right, so that duz dz is going to be zero at this boundary is the gradient condition being satisfied. We are defining the gradient of the velocity at this phase and at the other phases we are saying velocity itself is right. Okay. Now if we define uniform boundary uniform flow here which is for numerical solution which is possible we can we can simulate the developing region then this is here for a 2D case it is going to be a 2D flow here we can determine the developing length or entry length okay which is possible for numerical solution. Now with this problem statement we know the governing equation boundary conditions as set. Now let us try to get the analytical solution because you know as a first step when we are learning some numerical technique or some software in this case we are not getting to get we are not going to get into the numerical methods we will straight get into the software. So when we are learning some software it is better to have some analytical solution at hand so that we can compare the results. So we will start with the analytical solution. You know here I have written the z component of the momentum conservation equation I forgot to mention look at this momentum conservation equation vector equation so that means it has three components. I have written here we have written only the z component of the equation because other two components are not relevant. You see it is axisymmetric to a pipe it is an axisymmetric flow we can always assume and r momentum component there is no flow in the r direction so the r momentum component doesn't come into picture so the most relevant component is the z momentum component. z component of the momentum okay so assumptions are incompressible Newtonian flow a fluid given in the state problem statement steady state axisymmetric pipe flow laminar we have assumed so u z is only nonzero component is u z u r u and u theta 0 fully developed for analytical solution we have made this assumption. Constant transport properties that means row and u are constant and horizontal gz is zero we are not concentrating gravity mass conservation trivially satisfied and z component of momentum conservation we have written now we can see it in both ways one is because you know all because of all our assumptions. The left hand side of the equation is zero that means it is not an accelerating flow and if the flow is not accelerating the forces have to balance each other so there is this pressure force and there is this shear force. So forces have to balance each other and that is how I think even in any textbook have any textbook starts so they use they write these two terms right so you can relate to that. And when we write this this is then algebra and application of boundary conditions we get the form of the velocity profile with under all these assumptions. A similar exercise one can do for the channel. And I'm not see this is in the Cartesian coordinate system so the form of the equation changes however the principle remains exactly the same that forces have to balance for a non accelerating flow. And when the forces balance you we get the exact we get very similar equation and we solve it we get a slight we get a different form definitely here this RDR effect is not there. Okay, so we get set of equations for the channel. Now, flow through channel open for I think there is a tutorials hands on session after this and they will be taking it I will I will show you a demonstration for flow through pipe. Okay, so discretization when we take it when when we take the numerical rule then first we have to discretize the equation creation of geometry creation of mesh mesh assigning the boundary conditions selection of properties selection of solving. For setting the convergence criteria and analysis of results. Now, see when we talk about by we said it is axi-symmetric geometry. If it is axi-symmetric geometry, then do we really have to solve a 3d problem. What do you think. When problem is axi-symmetric sorry we have to solve when problem is axi-symmetric then we have to solve only one section and other section we will give same here. Absolutely, so in case and because it is finite volume method we have to there has to be a tiny volume we have to solve. So, here we can solve a wedge shaped geometry which is you know a delta theta right, conical shape geometry or a wedge shape geometry that's what we can solve. Right. Now, if you see in the in the in the file which you are now familiar with in a in a case file there are some basic directories right system. Constance and zero the initial condition right and in system we have the block mesh. In that block mesh day you we have to define our geometry for the simple thing right and here there we define the vertices. So, here you see we have defined it's a Cartesian coordinate system using the vertices we have created a geometry like this which is an wedge shaped geometry you know. This is my very poor effort of plotting this points and trying to show you the geometry. See the first point is 0000 and the second point is 0.4995 some number and 0.02181 and that is how this then the third point has been created the fourth point is here and then fifth and sixth point at that like that. The points are created there are nine points. Okay. Now, once we do the machine. This is the simple grading system. So, there is the see the the wall is in the in the radial direction. So, a grading in the mesh is there in the y direction in theta direction there is no that is only one mesh so scaling factories one in the art in the flow direction is that okay let it be uniform measure simple grading with this kind of mesh stretching the issue and these are the number of mesh number of reads in each direction. Okay, once the blocks are created block mesh is run we hear the boundary conditions inlet is patch outlet is patched and walls are created there is a bench for types of boundaries are assigned and then we have to run the block mesh once we run the block mesh you know if it runs successfully you get this kind of a you get this you get to see that you know ends successfully there is this end comment. Okay. After that you see this is the generation of the mission this is the way to check this and this is what I meant that there is a point five stretch ratio so the smallest size to the largest sizes point five and in the other direction this is the flow direction. Here it is one all of them are uniform and in the theta direction there is only one cell so is the geometry clear now. This is the art direction radial direction in the radial direction there is a tradition there is a expansion ratio near the wall this is the wall this is the wall near the wall this is the smallest mesh. Okay, near the center. This is the largest mesh smallest mesh to the largest mesh the stretch ratio is point five and that's what we have defined if you have seen we have defined one point five one this is that point five. This is the uniform mesh in the flow direction in a laminar flowing really don't expect much to happen so you know gradients to move smoothly rather so that's why in the uniform is in the flow direction it is uniform mesh and in the Z in the theta direction there is only one cell so this is also one this is how the mesh looks like. Right after the mesh we have the in the constant directory we have the transport properties transport properties you see we have to define only new. Here we have to we have to keep in mind in open form we don't we we we define only kinematic viscosity right right. That is new by room. Okay, what is the dimension. What is the dimension. If you use a sign meter square per second looking at this number which is the fluid I mean which is the fluid we are using here looking at this number. This is water water this is what okay so this is the new we have to specify. And after we have specified we have to give the velocity we have to define the boundary conditions. Here we have given the boundary conditions if this is the uniform we said we know in numerically we can define uniform bound. It is also so in the flow direction we said it is one zero two five zero zero other two components are zero outlet is zero gradient. So that means the use that is that is or do you X DX is zero walls are mostly so this is the boundary conditions we have supplied. Okay, now start time and time and you know I think down below somewhere convergence is there which I did not show you can take this control date file. Start time and time delta T is that these are the things to play around okay we can change we can change the end time we can change the delta T so that to take the time convergence and all this. After we run the simulation we have to go to Para view and here comes the analysis so that you know I have roughly about five minutes time to complete and in that time I will show you the analysis part. So first thing we have to plot the velocity as a function of radial position. Now where do we plot it see we know that we started with a uniform see this is the length of the pipe section right that way it's section and we know that we started with that uniform inlet condition and we expect the flow to develop right we expected the flow to develop. So, where should we plot it at the inlet condition or closer to the outlet condition or a different action location I would suggest that when we say start the. Exercise we plot a different actual position and see how the flow profile changes it start from a very uniform profile to a flat profile to a developed parabolic profile. Okay, please do this exercise I have plotted somewhere close to the exit and it looks like a parabolic profile now how do I know that I am getting a direct result see you remember we gave an uniform velocity profile which was 0.025 meter per second and here look the maximum velocity I'm getting somewhere close to point 0.05 so that is twice the average velocity and that's what one would expect in pipe flow. Okay, and then the other thing we have plotted is the center line velocity center line velocity what I have plotted against center line velocity I have plotted against the actual distance. Right, so you see center line velocity is increasing from 0.025 to 0.05 so now one can check where it is attaining the maximum velocity that is center line velocity where it is where it is attaining the maximum velocity that is 0.05 look at the graph and check what is the number where it is attaining the maximum velocity. You could get that actual position from the graph and you can calculate the Reynolds number see you know the average velocity which is 0.025 now 0.025 is the average is the average velocity diameter we know because we have specified the diameter here. Here we can see the half we can see the radius which is 0.005 so 0.005 is the radius so the diameter is 0.01 so 0.01 is the diameter 0.025 is the average velocity one can calculate the Reynolds number and tend to be power minus 6 is the new so what is the Reynolds number. Anybody calculating 250 is the Reynolds number right. Okay, if 250 is the Reynolds number what is going to be the entry length for laminar flow that entry length is L by D is equal to 0.06 le one could check this calculations from the note check go back to the textbook and check this calculations whether I am getting the correct result. And then the next thing with the developing length one can plot the pressure drop one is in the developing region pressure is going linear that means pressure drop is constant this is one check. The second check is pick up two points and put the pressure drop value put the pressure value this is calculate the delta P calculate the delta P from 32 we will be by this square this is another exercise and check whether we are getting the values close enough or not. So this is where one analyzes the result and cross checks if if it is not correct then goes back and what does the what do they do refine the mesh if it is laminar flow refine the mesh and if it is doubling for depending on the problem. They look at the problem set up problem formulation and with that I come to the end of the I think my time is also up. Yeah, discussion that low re flow through a pipe and channel is discussed implementation of low re flow through a pipe is the open form is discussed in analysis of results from simulation how do we analyze the result that has been discussed. And I acknowledge Mr. Ashley Melvin for creating the wage ship geometry. He was part of posse. And I thank all of you. Okay, if there are questions we'll just take it. How can we make wage ship geometry in open form. Okay, so let me go there. See these are this is the block mesh dict file. We have to. And this is how we have created the. This is the simple thing this is how we have created the vertices. Okay, and here we have used a scale factor of one zero one. Okay, you can use any other scale factor you can use any other numbers you can play around with this number. I can share this file with you so that you can play play around this number. So these are the vertices this is how we have created the vertices. And then what we have done is we then this text mesh is created that you know the vertex are joined to create the surfaces. Number of breeds in each block is specified in each direction is specified and what kind of reading is required that is specified here. So once this is done, then the boundaries are assigned. And this is how we created the wage ship mesh in the in open form. Okay, can you share this file. I will. Okay, thank you. I have a small doubt you said the pipe you are considering it as a circular pipe. And how is it gonna match the original dimension man like you are considering a wedge shape. But the pipe which are considering is a circle up one. So how does it give the accurate resolution for this kind of a measurement. I'm not able to. Okay, so do you have a piece of paper and a pen with you. Yes ma'am. Okay, so draw a circle. Yeah, okay. Okay, so at any and any angle. So this is actually right. Yeah. So at any angle draws some delta theta. Yeah, okay ma'am. So and this pipe you have you know this pipe you have the length and all that remains unaltered right. Yes ma'am. So if you instead of doing a full circle, you have to draw a circle. Yes ma'am. So if you instead of doing a full circle 3d pipe, if you just draw this wedge shaped thing. R is not altered. Length is not altered. Okay. It's just only this delta theta part. Okay ma'am. Okay ma'am. I get it. Okay. Thank you. Ma'am, there is one more question from Siddhi Vinayak. He's asking why you have used a sponsor ratio in radial direction and not in other direction. Sorry. Why you have used expansion ratio in radial direction but not in other direction. Okay. So see in radial direction because there is wall effect in the radial direction. Wall is there in the. Explain what is the wall effect. Okay. So see gradient direction here is the radial direction. Right. There is no gradient in the z direction or axial direction. That's why we have used uniform grid in the axial direction and stretch ratio in the wall direction. And finer grid close to the wall. What is the stretch ratio? Stretch ratio. See can you see this? Can you see my slide? See this is this is the radial direction. Okay. Yeah. This now see that difference that the size of this grid and the size of this grid. This is different, right? Yeah. So this smallest to the largest is the is the ratio that ratio we have used point five. That is I think they call the stretch ratio. Okay. Thank you. Ma'am one more question in the chat. Someone is asking to explain boundary condition for velocity and pressure. Okay. What we have done here is we have used the boundary condition as fixed value uniform boundary can uniform velocity at the inlet and zero gradient that is fully developed condition at the outlet. We did not use a pressure condition pressure is pressure will but pressure will get adjusted and no sleep at the wall. That is the boundary conditions we used. You can use a pressure outlet. See if you don't want to use a if you're unsure of its fully developedness and all you can use a pressure outlet condition but we did not use a pressure outlet condition. Hi madam. Could you explain once again why we have to use finer mesh in the wall near to the wall. See the gradient direction is the radial direction right. In this case flow direction there is of course in the developing region there is some gradient but beyond that we are setting the user dz to be zero. The fully developed region there is no gradient. If you take the derivative u z u r what do you get minus 2 r by r square. D u z d r is minus 2 r by r square. So again the gradient is zero at the center and maximum at the wall. So that means the changes in velocity with respect to the position is maximum at the wall. So that if you want to pick up you have to have finer grid there. Okay that's why we are using finer grid closer to the wall. Okay I'll tell you I'll give you one small thing. You have a piece of paper with you and pen. Yes madam. Draw a function somewhere it is flat and somewhere it is slightly steep. Slightly steeply varying. Okay. So where it is flat you can take a larger gap right and within the larger gap it is piecewise linear. Yes. But where it is sharply varying your gap has to be small. Your delta delta in the independent variable has to be small. To assume the dependent variable to be linear. Yeah the slope will be very high right that is what you are meaning. So that's what I'm saying here that you hear the variation of u z with respect to r is large near the wall as compared to it is near the center. So near the wall it is fine grid as compared to near the center. And now I understood madam thank you thank you so much. Yes. Can anyone tell me if there are any other questions in the chat. There is one more question from Roshan he is asking whether you can use mass flow inlet or mass flow outlet boundary condition. Mass flow inlet is possible. Mass flow outlet he can check whether he is getting the fully developed I need to check before I say yeah. Yeah. For this kind of situation I always was fully developed it should develop it should develop. Madam just one general question. Yeah. The finer mesh at the wall is this will it be this case for almost like all the general flows. Yes. Yeah because on almost all example like whenever we are like the professors were sharing the screen like it was observed that near to the wall the mess just will be always finer. Yes. Yes that you can like you know this is one thing you can you always want to capture the wall effect. Okay. So is there any other question like you know is there anything in the chat. Wall effect means wall shear stress man. Wall shear stress in case of you know see at the wall what happens is velocities becomes zero right and then the gradient is high so that you want to resolve. Okay ma'am Satya Narayan is asking about the boundary condition that you have implemented in axial direction. Sorry. Boundary condition that was implemented in axial direction. Boundary conditions in the actual direction yes. Yes ma'am. Also at the wall like no sleep at the wall. Yeah yeah also at the wall in radial and axial direction is asking like. Every face we have to give the boundary condition. Ma'am again a follow up question. So this wall effect is it something generally people just refer in papers like y plus is equal to something one or something like that. That is for turbulent flow. Okay. I have one more question here. Sometime in outlet we use pressure gradient and sometime we use zero pressure I mean a constant pressure. So is there any specific conditions where we should be choosing out of these two at the outlet that when we should use zero pressure gradient or when we should use constant pressure boundary condition at the outlet. Zero velocity you mean zero velocity gradient or constant pressure. No no no it outlet sometime we use zero pressure gradient and sometime we use a pressure constant. So is there any specific criteria. Pressure gradient you use zero pressure. Dopey by doing equal to zero at outlet sometime we use like that. Or if I'm wrong please correct me. Okay zero pressure gradient is something I need to check. Okay constant pressure you can most of in the pressure outlet condition is the most general thing you don't really have to assume anything. Okay if you're when you are completely sure that your flow will be fully developed you can use a zero velocity gradient. Zero pressure gradient is something I'll just check. Unless there is a velocity driven flow I'm not sure where you are using a zero pressure gradient I'll just check at the outlet. Okay. Yeah for example let's say example of a flapping wing. So if you're estimating such kind of a flow you will have zero pressure gradient even at the outlet sometime you use that. Even the use a convection boundary condition. Sorry ma'am. It's problem dependent I'll have a look at the problem if I look at the problem I'll be able to tell you. So boundary conditions are essentially problem dependent. Yeah so I was just wondering that is there any criteria where we can choose out of these two that okay at this time we'll go through pressure constant or maybe next time I can go through zero pressure gradient something like this. So is there any criteria. We have to look at the problem. Yeah yeah thank you. Thank you bro. Ma'am one more question. So when you're simulating this flow through a pipe at the inlet generally it is we prescribe the velocity and the pressure is prescribed as del p is equal to zero and outlet we say the del of u is equal to zero and p is equal to some p outlet. So is there any reason why it is like that generally I mean why we don't prescribe the pressure at the inlet instead of the. See if you define the pressure at the. Typically in general what happens is when we simulate something we simulate the practical problem. And we know the in any experiment you can think of we know the flow rate. And that's why we define the flow rate at the inlet either the mass flow rate or the uniform average velocity or something right. That's what we define. And at the outlet what happens is we do not know the exact velocity profile. Right. So sometimes. Yeah this is with respect to the pressure actually so. I'll tell you. So typically do not know the exact velocity profile so somebody said that somebody mentioned that can we use a mass. Mass flow out outlet mass flow condition that is one way sometimes and most of the times we know the pressure. So we can give a pressure pressure outlet condition if you give the pressure outlet condition it and a velocity inlet condition or mass inlet condition pressure at the inlet gets adjusted you don't have to we don't have to define the inlet pressure again. Okay. So only if a pressure driven flow we don't have to define the inlet pressure again it gets adjusted it gets adjusted that's why we don't define the pressure inlet if we define. We don't have to define the inlet velocity. Yeah that is my confusion like you so you're told that the pressure gets adjusted like I do not understand that particular point. I'll just. I'll show you from the laminar flow right using laminar flow that is easy. So see this this equation see pressure gradient is related to you max and you max is of course it's the you max is related to the are the velocity profile. Is related to you max is here so pressure pressure and pressure gradient and the velocity profiles are related to each other. So now if we know the inlet velocity profile and the outlet pressure the inlet pressure is this gets calculated. If you look at this. So using R is what minus partial P partial Z R squared by four new one minus R squared by my smaller square by capital R squared. Okay and this if we say that and this is linear same case of fully developed I'm just saying this this is linear so I'll write it delta P by L. So delta P is the inlet minus P outlet. So if I know P outlet and if I know inlet velocity profile inlet velocity P inlet gets calculated and that's what I. Okay. Okay ma'am. So we don't have to give the inlet. See flow will happen because of the pressure gradient flow we know outlet pressure we know inlet pressure gets calculated. Right. Hello ma'am. Hello. Get any error if we prescribe the pressure at the inlet like interchange like we instead of prescribing the pressure at the outlet along with you inlet if we I mean this is general not with respect to this example I am asking but generally. In general I think if I mean so outlet you have to give a boundary condition right. Yes yeah so generally it is mentioned that at the inlet you prescribe the velocity and at the outlet you prescribe the pressure. That is the most general boundary condition. Yeah so I was trying to figure out the like answer for that like why is it so like instead can we also why can't we prescribe the pressure also at the inlet like velocity. So if you see you know if you define pressure inlet and velocity outlet. Yeah like say have the same boundary condition for the like you have the U inlet and delta U is equal to 0 at outlet. Same thing for pressure also like P inlet and delta P is equal to 0. I think that will be over specified you can try I mean I do not know but I think one of some of the one of the boundary conditions or something will be overridden that will be there within the code. You can try that just try and see what happens. Okay. See if you see if you define both pressure and velocity at both inlet and outlet I think it is it is going to be over specification. Yeah what I was meaning was at the inlet you say P is equal to the inlet pressure and outlet you specify the gradient delta P is equal to 0. Delta P is equal to 0. You can say delta P between what like the partial P partial Z is equal to 0. I am saying zero gradient at the outlet generally it is the other way around the pressure. I actually see you cannot have the Z see if it is a pressure driven flow you cannot have the I can't see why how you can have a I can't see how you can have a delta P delta Z 0. Okay so that has to be a pressure gradient rate for the flow to flow to happen. If it is a pressure driven flow that has to be there has to be a positive pressure gradient for the flow to happen right. I mean negative pressure gradient positive delta P for the pressure to happen for the flow to happen right. So I do not see for a pressure driven flow how you will set the delta P delta Z 0. But what you could try to simulate you could try and see what happens in any in different solvers defining pressure inlet and pressure outlet condition and see how the flow develops. For example suppose if you have already developed flow you know the inlet you outlet and all these things you know the pressure at inlet. Give that same pressure conditions and see how the flow develops that should develop just check that you can check that run a simulation. Sure madam thank you so much. For a pressure driven flow I do not see how you can give the delta P delta Z 0 at outlet that I do not think it should work. Hello ma'am thank you good morning and thank you for your lecture. Ma'am I have been one doubt like already in the past I have seen in the flow and that for boundary condition they have mentioned that if we are generating a constant density flow in compressible flow then in that case velocity inlet and pressure outlet is the most convenient boundary condition. That is the most common. For this for ink for compressible flow they prefer mass flow rate inlet at inlet boundary condition and at outlet mass flow rate outlet boundary condition. So here I am getting confusion from the basic conservation principle whatever the flow it's may be a incompressible compressible already developing or fully developed flow this mass flow rate must be conjured at both of the. And outlet then why cannot be used and I have tried this mass flow rate inlet and mass flow rate outlet at boundary condition for in compressible flow and I was having a solution which is having a kind of deviation I were not getting a correct solution. So I am having this doubt. So that is very specific like no that is a very specific numerical question we need to check that we need to check which algorithm you are using and all. Ma'am like whatever it can be. Your question is valid you should it should work we need to check that. Okay. Yeah okay ma'am thank you. Okay so I think we can end the session. Professor Marswita. Yeah I think. Yeah okay okay thank you so much. It was such a wonderful session and so many quick varies and very much informative. Thanks.